A Concise Course in Algebraic Topology

Algebraic topology is a basic part of modern mathematics, and some knowledge of this area is indispensable for any advanced work relating to geometry, including topology itself, differential geometry, algebraic geometry, and Lie groups. This book provides a detailed treatment of algebraic topology both for teachers of the subject and for advanced graduate students in mathematics either specializing in this area or continuing on to other fields.J. Peter May's approach reflects the enormous internal developments within algebraic topology over the past several decades, most of which are largely unknown to mathematicians in other fields. But he also retains the classical presentations of various topics where appropriate. Most chapters end with problems that further explore and refine the concepts presented. The final four chapters provide sketches of substantial areas of algebraic topology that are normally omitted from introductory texts, and the book concludes with a list of suggested readings for those interested in delving further into the field. Introduction Chapter 1. The fundamental group and some of its applications 1. What is algebraic topology? 2. The fundamental group 3. Dependence on the basepoint 4. Homotopy invariance 5. Calculations: π1(R) = 0 and π1(S1) = Z 6. The Brouwer fixed point theorem 7. The fundamental theorem of algebra Chapter 2. Categorical language and the van Kampen theorem 1. Categories 2. Functors 3. Natural transformations 4. Homotopy categories and homotopy equivalences 5. The fundamental groupoid 6. Limits and colimits 7. The van Kampen theorem 8. Examples of the van Kampen theorem Chapter 3. Covering spaces 1. The definition of covering spaces 2. The unique path lifting property 3. Coverings of groupoids 4. Group actions and orbit categories 5. The classification of coverings of groupoids 6. The construction of coverings of groupoids 7. The classification of coverings of spaces 8. The construction of coverings of spaces Chapter 4. Graphs 1. The definition of graphs 2. Edge paths and trees 3. The homotopy types of graphs 4. Covers of graphs and Euler characteristics 5. Applications to groups Chapter 5. Compactly generated spaces 1. The definition of compactly generated spaces 2. The category of compactly generated spaces Chapter 6. Cofibrations 1. The definition of cofibrations 2. Mapping cylinders and cofibrations 3. Replacing maps by cofibrations 4. A criterion for a map to be a cofibration 5. Cofiber homotopy equivalence Chapter 7. Fibrations 1. The definition of fibrations 2. Path lifting functions and fibrations 3. Replacing maps by fibrations 4. A criterion for a map to be a fibration 5. Fiber homotopy equivalence 6. Change of fiber Chapter 8. Based cofiber and fiber sequences 1. Based homotopy classes of maps 2. Cones, suspensions, paths, loops 3. Based cofibrations 4. Cofiber sequences 5. Based fibrations 6. Fiber sequences 7. Connections between cofiber and fiber sequences Chapter 9. Higher homotopy groups 1. The definition of homotopy groups 2. Long exact sequences associated to pairs 3. Long exact sequences associated to fibrations 4. A few calculations 5. Change of basepoint 6. n-Equivalences, weak equivalences, and a technical lemma Chapter 10. CW complexes 1. The definition and some examples of CW complexes 2. Some constructions on CW complexes 3. HELP and the Whitehead theorem 4. The cellular approximation theorem 5. Approximation of spaces by CW complexes 6. Approximation of pairs by CW pairs 7. Approximation of excisive triads by CW triads Chapter 11. The homotopy excision and suspension theorems 1. Statement of the homotopy excision theorem 2. The Freudenthal suspension theorem 3. Proof of the homotopy excision theorem Chapter 12. A little homological algebra 1. Chain complexes 2. Maps and homotopies of maps of chain complexes 3. Tensor products of chain complexes 4. Short and long exact sequences Chapter 13. Axiomatic and cellular homology theory 1. Axioms for homology 2. Cellular homology 3. Verification of the axioms 4. The cellular chains of products 5. Some examples: T , K, and RPn Chapter 14. Derivations of properties from the axioms 1. Reduced homology; based versus unbased spaces 2. Cofibrations and the homology of pairs 3. Suspension and the long exact sequence of pairs 4. Axioms for reduced homology 5. Mayer-Vietoris sequences 6. The homology of colimits Chapter 15. The Hurewicz and uniqueness theorems 1. The Hurewicz theorem 2. The uniqueness of the homology of CW complexes Chapter 16. Singular homology theory 1. The singular chain complex 2. Geometric realization 3. Proofs of the theorems 4. Simplicial objects in algebraic topology 5. Classifying spaces and K(π, n)s Chapter 17. Some more homological algebra 1. Universal coefficients in homology 2. The Künneth theorem 3. Hom functors and universal coefficients in cohomology 4. Proof of the universal coefficient theorem 5. Relations between ⊗ and Hom Chapter 18. Axiomatic and cellular cohomology theory 1. Axioms for cohomology 2. Cellular and singular cohomology 3. Cup products in cohomology 4. An example: RPn and the Borsuk-Ulam theorem 5. Obstruction theory Chapter 19. Derivations of properties from the axioms 1. Reduced cohomology groups and their properties 2. Axioms for reduced cohomology 3. Mayer-Vietoris sequences in cohomology 4. Lim1 and the cohomology of colimits 5. The uniqueness of the cohomology of CW complexes Chapter 20. The Poincaré duality theorem 1. Statement of the theorem 2. The definition of the cap product 3. Orientations and fundamental classes 4. The proof of the vanishing theorem 5. The proof of the Poincaré duality theorem 6. The orientation cover Chapter 21. The index of manifolds; manifolds with boundary 1. The Euler characteristic of compact manifolds 2. The index of compact oriented manifolds 3. Manifolds with boundary 4. Poincaré duality for manifolds with boundary 5. The index of manifolds that are boundaries Chapter 22. Homology, cohomology, and K(π, n)s 1. K(π, n)s and homology 2. K(π, n)s and cohomology 3. Cup and cap products 4. Postnikov systems 5. Cohomology operations Chapter 23. Characteristic classes of vector bundles 1. The classification of vector bundles 2. Characteristic classes for vector bundles 3. Stiefel-Whitney classes of manifolds 4. Characteristic numbers of manifolds 5. Thom spaces and the Thom isomorphism theorem 6. The construction of the Stiefel-Whitney classes 7. Chern, Pontryagin, and Euler classes 8. A glimpse at the general theory Chapter 24. An introduction to K-theory 1. The definition of K-theory 2. The Bott periodicity theorem 3. The splitting principle and the Thom isomorphism 4. The Chern character; almost complex structures on spheres 5. The Adams operations 6. The Hopf invariant one problem and its applications Chapter 25. An introduction to cobordism 1. The cobordism groups of smooth closed manifolds 2. Sketch proof that N_∗ is isomorphic to π_∗(TO) 3. Prespectra and the algebra H_∗(TO; Z2) 4. The Steenrod algebra and its coaction on H_∗(TO) 5. The relationship to Stiefel-Whitney numbers 6. Spectra and the computation of π_∗(TO) = π_∗(MO) 7. An introduction to the stable category Suggestions for further reading 1. A classic book and historical references 2. Textbooks in algebraic topology and homotopy theory 3. Books on CW complexes 4. Differential forms and Morse theory 5. Equivariant algebraic topology 6. Category theory and homological algebra 7. Simplicial sets in algebraic topology 8. The Serre spectral sequence and Serre class theory 9. The Eilenberg-Moore spectral sequence 10. Cohomology operations 11. Vector bundles 12. Characteristic classes 13. K-theory 14. Hopf algebras; the Steenrod algebra, Adams spectral sequence 15. Cobordism 16. Generalized homology theory and stable homotopy theory 17. Quillen model categories 18. Localization and completion; rational homotopy theory 19. Infinite loop space theory 20. Complex cobordism and stable homotopy theory 21. Follow-ups to this book